On 2013-02-19, Dave <divergent.tseries@gmail.com> wrote:> I sadly work with the Cauchy distribution all the time. It is the natural distribution function for the data I work with, although it is part of a mixture distribution.

> People regularly mistake Cauchy driven data for normally driven dataas people do not realize how common it is nature. Poisson's commentto Laplace about only needing to footnote the existence of the Cauchydistribution as it will not be encountered in practice hasn't helped any.

> One thing that Bayesian statistics lacks, that Frequentists have in abundance is a nice, pre-built set of diagnostic tools.

The alchemists and other medieval "natural philosophers" likewisehad such tools. As with the case then, these tools are often verywrong. The situation is not as simple.

One which really needs to be eliminated is "statistical sibnificance".There ARE cases where it makes some sense, but the point null hypothesisas stated is always false, and the real question is what action shouldbe taken. A statistically significant difference may be of no practicalor even theoretical importance, and a statistically insignificant differencemay be very important. This simple tool is like a loaded gun in the handsof an ignoramus as to how guns work.

> I am philosophically neutral on the Bayesian/Fisherian/Neyman-Pearsonissue. I am a deep believer in doing what works. I also have a strongpreference for things like the t-test (properly used) as it requires nothinking and no work. Cost-benefit should never be ignored.

Cost-benefit should not be ignored, and in fact the basis of decisiontheory is that of comparing risks. Simple axioms of self-consistentbehavior, see for example my paper is _Statistics and Decisions_ 1987,show that one must compare integrated risks. IF one has a good ideaof the integrating factor, the optimal solution is Bayesian, againIF it can be computed. Those are big ifs, and approximations canbe made.

> I do think that part of the genius of Frequentist statistics is itsdecision tree or algorithmic nature. If you encounter data like this,then analyze it like that. If some different assumption is presentor a warning statistic clicks on as a problem then do some other thinginstead as a robustness check.

There is nothing here which is foreign to decision theory, definitelyincluding Bayesian.

> But a Frequentist is presuming to know the true model of the worldand so has to be able to do this. A Bayesian is inducing the model fromthe data. Still, it wouldn't kill us to have a decision tree like...ifyour problem looks like this then the following likelihood functionscould work to model the problem. And further, the following posteriorsimulations do X,Y,Z in practice. Except for the conjugate families,we don't really have this.

Your statements are incorrerct. Classical philosophy of sciencehas the scientist deriving the theory from data. This is false;one can only choose between theories from the data. The idealdecision make has all models in his mind and chooses between them;we are not ideal, so we have to put some weight on "something else".Again, this means that we imperfect decision makers have to use prior Bayes methods which allow for such errors.

-- This address is for information only. I do not claim that these viewsare those of the Statistics Department or of Purdue University.Herman Rubin, Department of Statistics, Purdue Universityhrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558